You might want to look into a Kawasaki Ninja 250, or if youre lucky you might find an EX500. I doubt for under a grand though.

Another bike in your range, and something "sporty" (as sporty as it gets for $1k) is a Suzuki GS500e. That'll do the trick

If you can scrounge together about 3k you can get yourself a ~2000 Suzuki SV650s and a little less will get you an SV650.

The SV(or s) is an amazing handling motorcycle, though you can't adjust the front forks. For your needs, and 95% of the world, it's more than adequate in it's stock form, for handling.

Those are some suggestions.